Properties Of S-tensors

Tensor complementarity is a new topic in tensor research.On the one hand,tensor complementarity problem is an extension of linear complementarity problem under tensor(supermatrix).Many of its theoretical results are the extension of theoretical research on linear complementarity problem.But the reason why there are more handicap to resolve the tensor complementarity problem is that we have to figure out lots of problem arise from the nonlinearity of polynomials.On the other hand,tensor complementarity problem is a special subcategory of the nonlinear complementarity problem.According to the structure of tensor and the special properties of corresponding polynomials,we already have some new results which can not be found in the research of nonlinear complementarity problem.In recent years,more and more scholars put much emphasis on the structural tensor which is the important fundament of the research of the tensor complementarity problem.Many feasibility of tensor complementarity problem,stability,existence,global uniqueness and boundedness to the solution of tensor complementarity are based on the special properties of the relevant structural tensor.Especially,in the theoretical and algorithmic research of tensor complementarity problem,the(strict)feasibility is a basic and important problem.The strict feasibility of tensor complementarity problem can be described by S tensors.Therefore,the study of strictly feasibility of the tensor complementarity problem could be transformed into a study of a special structural tensor – S-tensors.With the wider research of the practical application of the tensor complementarity problem.The higher request for solving and calculating the tensor complementarity problem are proposed on the basis of relevant theories.As one of the important algorithm for solving complementarity problems,the feasible interior point algorithm have to satisfy the problem which we solved is feasible.Therefore,the study of S tensor in this paper provides an important theoretical basis to designing a feasible interior point algorithm for solving tensor complementarity problems.In this paper,we discuss properties of S-tensors.We illustrate the principal subtensors of an S tensor are not always the S-tensor.In particular,we establish several necessary and/or sufficient conditions to judge whether a tensor is an S-tensor or not.Moreover,we also discuss relationship among S-tensors and several related tensors.